# Unique prime

In mathematics, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique iff there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q. Unique primes were first described by Samuel Yates in 1980.

It can be shown that a prime p is of unique period n iff there exists a natural number c such that

[itex]\frac{\Phi_n(10)}{\gcd(\Phi_n(10),n)} = p^c[itex]

where Φn(x) is the n-th cyclotomic polynomial; until today, 18 unique primes are known, and no others exist below 1050. The following table gives an overview of all known unique primes Template:OEIS and their periods Template:OEIS:

Period lengthPrime
13
211
337
4101
109,091
129,901
9333,667
14909,091
2499,990,001
36999,999,000,001
489,999,999,900,000,001
38909,090,909,090,909,091
191,111,111,111,111,111,111
2311,111,111,111,111,111,111,111
39900,900,900,900,990,990,990,991
62909,090,909,090,909,090,909,090,909,091
120100,009,999,999,899,989,999,000,000,010,001
15010,000,099,999,999,989,999,899,999,000,000,000,100,001

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